STRONG LAWS OF LARGE NUMBERS FOR ARRAYS OF ROW-WISE EXCHANGEABLE RANDOM ELEMENTS

Let {Xnk _< k _< n, n _> 1} be a triangular array of row-wise exchangeable -1/ random elements in a separable Banach space. The almost sure convergenceof, =1 Xnk, ! p < 2, is obtained under varying moment and distribution conditions on the random elements. In particular, strong laws of large numbers follow for triangular arrays of random elements in (Rademacher) type p separable Banach spaces. Consistency of the kernel density estimates can be obtained in this setting.

1. INTRODUCTION AND PRELIMINARIES.Blum et al. [1] obtained central limit theorems for arrays of exchangeable random variables using a version of de Finetti's theorem which implied that an infi- nite sequence of exchangeable random variables is a mixture of sequences of indepen- dent, identically distributed random variables.Taylor [2] used similar techniques in obtaining weak and strong convergence results for arrays of random elements which are row-wise exchangeable.Using martingale methods, Weber [3] developed central limit results for triangular arrays of random variables which were row-wise exchange- able.tlis methods did not require infinite exchangeability or the de Finetti representation.In this paper, almost sure convergence is obtained for --1Z= Xnk and n -1/p n n Ek= Xnk in separable Banach spaces using martingale methods.These results are for triangular arrays, and hence only finite exchangeability in each row is re- quired.By assuming convergence in mean for each column, the hypothesis of the pre- viously cited limit theorems are substantially relaxed.
Let E denote a real separable Banach space with norm II II.Let (, A, P) denote a probability space.A random element X in E is a function from into E which is A-measurable with respect to the Borel subsets of E, 8(E).The pth absolute moment of a random element X is EIIXII p where E is the expected value of the (real-valued) random variable IXI p.The expected value of X is defined to bc the Bochner integral (when E 11XII < ) and is denoted by EX.The concepts of independence and identical distributions (i.i.d.) have direct extensions to E.
shows that all joint probability laws are the same and that exchangeable random elements are identically distributed.Finally, a subset B of E whose boundary DB satisfies

P(B)
0 is called a P-continuity set.
2. STRONG LAWS OF LARGE NUMBERS FOR TRIANGULAR ARRAYS.
The main result of this section is a strong law of large numbers.Moment con- ditions and a measure of nonorthogonality condition will be assumed on the distribu- tions of the random elements.Throughout this section {Xnk" _< k _< n, n _> i} will denote an array of random elements in a separable Banach space E which are row-wise exchangeable.
First, two preliminary results will be presented for later use in Sections 2 and 3.The first result shows that the infinite sequence, formed by the convergence in rth mean of each column of the triangular array, is exchangeable when each row consists of exchangeable random elements.This allows the application of a version of de Finetti's theorem to the limit sequence.
LEIA I. Let {Xnk _< k _< n, n _> i} be an array of random elements which are row-wise exchangeable.If the random elements converge in the rth mean to Xook(r > O) for each k, then the sequence {Xoo k" k > i} is exchangeable.A k which are PXl-continuity sets.
Since the limits are unique, and the PX -continuity sets form a determining lass, it follows that l' for all (B B k) e 8(Ek).Thus, (Xoo Xok) and (XI Xok) have identical joint distributions, tlence, the sequence (Xo k" k >_ 1} is exchangeable./// REMARK.Note that the convergence of the joint distributions is sufficient in the proof of Lemma 1.Unfortunately, this is not implied by convergence in distri- bution in each column.
For arrays where each row is an infinite sequence of exchangeable random elements, Olshen [5] showed that de Finetti's theorem implied that for each n P(Bn ; P(Bn)dn (Pv)   (2.1) where F denotes the collection of probabilities on the Borel subsets of E and P (B) n E m is the probability of Bn [g(Xnl Xnm) Bn] (where g E is a Borel function) computed under the assumption that {Xnk" k >_ l} are independent, identically distri- buted random elements and V is the mixing measure defined on 8(F).The next result n shows that if {Xnk" _< k _< n, n _> l} are row-wise exchangeable random elements which converge in the second mean for each k and E[f(Xnl f(Xn2)] 0 as n then E[f(Xl) f(X2) 0 and Ev(Xol) 0 where E v is the expectation with respect to P Moreover, it also follows that EX 0.
LEMMA 2. Let {Xnk" _< k _< n, n _> I} be an array of row-wise exchangeable random elements in a separable Banach space such that {Xnk} converges in the second mean to Xk for each k.If for each f E* n Ev(XI) 0 for Pv with U-probability one, and (iii) E(XI) O.   2  for each k it is clear that PROOF.Since Xnk Xk sup El]nkll 2 < and EIIXkl] 2 < n for each k.Using (2.2) and convergence in the 2nd mean, for each which goes to 0 as n by hypothesis (i).Thus, xallllxll)] Since convergence in the second mean implies convergence in mean, it follows from Lemma that the sequence {Xk k > l} is exchangeable.Also it follows from (2 i) which implies that Evf(Xl) 0 p a.s.for each f E*.Since the Bochner integral implies the Pettis integral in a separable space, f(Ev(Xool)) E(f(Xool) 0 oo a.s.
for each f which implies that Ev(Xool) 0 Voo a s by the Hahn-Banach theorem and the separability of E. Also, EXI f F Ev(XI) d(Pv) O.
/// that the conclusions of Lemma 2 do not necessarily imply that {Xk k > 1} is a sequence of i.i.
THEOREM I. Let {Xnk" _< k _< n, n _> i} be an array of random elements in a separable Banach space.Let {Xnk} be row-wise exchangeable fo-each n and let Unn and Uoon be the o-fields defined in (2.4In this section, strong laws of large numbers for triangular arrays of row-wise exchangeable random elements in type p 8 separable Banach spaces will be establish- ed.Recall that a separable Banach space is said to be of type p, p 2, if there exists a constant C such that n Xkl p < C n El iXkl ip El IEk: Ek= for all independent random elements I' Xn with zero means and finite pth moments.Every separable Banach space is type I.The next result by Woyczynski [7] for sequences of zero mean, independent random elements in E with uniformly bounded tail probabilities is listed for future reference.
TtlEOREM 2. Let < p < 2. The following properties of a Banach space E are equivalent" (i) For any sequence {X.} of zero mean, independent random elements in E wtth lip uniformly bounded tail probabilities the erie verges a.s.
(iii) For any sequence {X.} as in (ii) 1/p n xll Oa lln-Ek= Since (Xk k > l} are exchangeable they are identically distributed tlence they have uniformly bounded tail probabilities.That is, for all t g R and k 1, llxll > t) < gllXlll > t).Tus a strong convergence result for row-wise exchangeable in type p separable Banach spaces can be obtained using Theorem 2 and the techniques of Theorem 1. THEOREM 3. Let (Xnk" _< k _< n, n _> I} be an array of random elements in a separable Banach space of type p 6, <_ p < 2. Let {Xnk} be exchangeable for each n and let U and U be the o-fields defined in (2.4).If nn on (i) [IXnl Xl]l IX(n+l), Xooll for each n, (ii) E IXnl Xok]l 2 o(n-2a), where a (p-1)/p, and (iii) pn(f) E[f(Xnl)f(Xn2)] 0 as n for each f g E*, then ii1/nl/p n II 0 a s Ek=l Xnk PROOF" Since Ell Xnl Xoo II 2   2   o(n-2e), then Xnk Xk.By Lemma 1, (Xk" k 1} is an infinite exchangeable sequence of random elements.where {Xk k _> 1} are independent and identically distributed with respect to Pv.

1/PF.= 1Xok
By Lemma 2, Ev(Xook) 0. Thus, by Theorem 2 (Theorem for p 1) In-  REMARK.It should be noted that if lXnl xlll 0 a.s., then the condition I[Xnl col[ IIX(n+l), Xcol[ for all n is not needed in the proof of Theorem 3.That is lXnl Xml II 0 a.s.implies that IE(Xnl Xco IUoon) II 0 a.s.which is crucial to the proof of Theorem 3. It is sometimes easier to show directly that ICcxt-Xn) lUn)l 0 a.s. as will be demonstrated in the kernel density esti- mation example to follow.
The following example considers the general density estimation problem where X,, X are independent and identically distributed random variables with the same density function f.EYuMPLE.Let X X be independent identically distributed random varia- bles with common density function f.The kernel estimate for f with constant bandwidths h is given by n where K is often chosen to be a bounded (integrable) kernel with compact support [a,b] and h 0 an n co.For additional background material see l)eVrove n and Wagner [8] and Taylor [9].Let E {gig" R R and Thus, E is a separable Banach space of type min{2,p}.Define t-Xk t-Xl Xnk --(K(--ff---) E[K(--ff)]) (3.1)   n n n It is clear that Xnk E and that {Xnk} is independent and henceexchangeable for each n.The next proposition will prove directly that [IE(^nl [Umn)[ 0 a.s.
In this setting XI 0 a.s.

II!
Depending on the choice of K, IlXnlll may not converge to 0 Xl in the example).Also, Theorem 3 and the example emphasizes the importance of p being as large as possible (< 2).Moreover, R, Rm, ltilbert spaces and all finite-dimen- sional Banach spaces are of type 2, and consequently they are type p for each p<2.Finally, it is important to observe that the resuIts are substantial new results even for real-valued random variables.
. which implies that Pv[supl In r.k= Xk ] 0 as m .By the bounded n>m convergence theorem, 1/p n g f F p[supl In-gk= x k > 1 d(Pv) 0 as ra oo.
d, random elements.The final result of this section is a strong law of large numbers for triangular arrays of random elements which are row-wise exchangeable.Define n Zk= Xnk E(Xnl[Unn) E(Xnl]Um) a.s. and n =E as Zk=l Xook (Xcol Uon)( ).Let {Xnk} converge in the second mean E P[supl IE(XnllUoon) F.(Xoo lu)l > 1 n>m n c p[supl I Ek= Xookl > g] n>m P[supl IE(Xnl Xoo lu)l > 1